The manifold hypothesis posits that for many problems, your data samples lie on a low-dimensional manifold embedded in the original encoding space.
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Being able to fit a curve that approximates the latent space relies critically on two factors: 1. The structure of the latent space itself! (a property of the data, not of your model) 2. The availability of a "sufficiently dense" sampling of the latent manifold, i.e. enough data
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You *cannot* generalize in this way to a problem where the manifold hypothesis does not apply (i.e. a true discrete problem, like finding prime numbers).
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In this case, there is no latent manifold to fit to, which means that your curve (i.e. deep learning model) will simply memorize the data -- interpolated points on the curve will be meaningless. Your model will be a very inefficient hashtable that embeds your discrete space.
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The second point -- training data density -- is equally important. You will naturally only be able to train on a very space sampling *of the encoding space*, but you need to *densely cover the latent space*.
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It's only with a sufficiently dense sampling of the latent manifold that it becomes possible to make sense of new inputs by interpolating between past training inputs without having to leverage additional priors.pic.twitter.com/SmRvEN2NXS
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The practical implication is that the best way to improve a deep learning model is to get more data or better data (overly noisy / inaccurate data will hurt generalization). A denser coverage of the latent manifold leads a model that generalizes better.
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This is why *data augmentation techniques* like exposing a model to variations in image brightness or rotation angle is an extremely effective way to improve test-time performance. Data augmentation is all about densifying your latent space coverage (by leveraging visual priors).
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In conclusion: the only things you'll find in a DL model is what you put into it: the priors encoded in its architecture and the data it was trained on. DL models are not magic. They're big curves that fit their training samples, with some constraints on their structure.
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End of conversation
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